At any node/vertex the sum of angles is 420 degrees,an extra equilateral triangle ( any side is of unit length) is pushed in. This forms a warped polyhedral surface. (Six triangles can be assembled to make a flat hexagon).

Not able to further find coordinates for other points in the triangulation/tessellation.

My main motivation in starting this is that.. so many symmetric tesselations around origin for elliptic geometry exist but ( as much as I know ) none in hyperbolic geometry except those forming on surfaces of revolution...